using UnPack
using Plots
using Random
using ComponentArrays
using BenchmarkTools
Random.seed!(0)
function mymodel(u0, p, t)
T = eltype(u0)
@unpack r, K = p
@. K / (one(T) + (K - u0) / u0 * exp(-r * t))
end
p = ComponentArray(;r = 1., K = 1.)
u0 = 0.005
tsteps = range(0, 20, length=100)
y = mymodel(u0, p, tsteps)
plot(tsteps, y)A primer on mechanistic inference with differentiable process-based models in Julia
Here you will learn about different techniques to infer parameters of a (differentiable) process-based model against data. This is useful to in the context of mechanistic inference, where we want to explain patterns in a system by understanding the processes that generate them, in contrast to purely statistical or empirical inference, which might identify patterns or correlations in data without necessarily understanding the causes. We’ll mostly focus on differential equation models. Make sure that you stick to the end, where we’ll see how we can not only infer parameter values but also functional forms, by parametrizing models’ components with neural networks.
Preliminaries
Wait, what is a differentiable model?
One can usually write a model as a map \(\mathcal{M}\) mapping some parameters \(p\), an initial state \(u_0\) and a time \(t\) to a future state \(u_t\)
\[u_t = \mathcal{M}(u_0, t, p).\]
We call differentiable a model \(\mathcal{M}\) for which we can calculate its derivative with respect to \(p\) or \(u_0\). The derivative \(\frac{\partial \mathcal{M}}{\partial \theta}\) expresses how much the model output changes with respect to a small change in \(\theta\).
Recall your Calculus class!
\[\frac{df}{dx}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
Let’s illustrate this concept with the logistic equation model. This model has an analytic formulation given by:
\[\mathcal{M}(u_0, p, t) = \frac{K}{1 + \big( \frac{K-u_0}{u_0} \big) e^{rt}}\]
Let’s code it
What is a
ComponentArray?
A ComponentArray is a convenient Array type that allows to access array elements with symbols, similarly to a NamedTuple, while behaving like a standard array. For instance, you could do something like
cv = ComponentVector(;a = 1, b = 2)
cv .= [3, 4]ComponentVector{Int64}(a = 3, b = 4)
This is useful, because you can only calculate a gradient w.r.t a Vector!
Now let’s try to calculate the gradient of this model. While you could in this case derive the gradient analytically, an analytic derivation is generally tricky with complex models. And what about models that can only be simulated numerically, with no analytic expressions? We need to find a more automatized way to calculate gradients.
How about the finite difference method?
The gradient of the model is useful to understand how a parameter influences the output of the model. Let’s calculate the importance of the carrying capacity K on the model output:
dm_dp = ∂mymodel_∂K(1e-1, u0, p, tsteps)
plot(tsteps, dm_dp)As you can observe, the carrying capacity has no effect at small \(t\) where population is small, and its influence on the dynamics grows as the population grows. We expect the reverse effect for \(r\).
On the importance of gradients for inference
The ability to calculate the derivative of a model is crucial when it comes to inference. Both within a full Bayesian inference context, where one wants to sample the posterior distribution of parameters \(\theta\) given data \(u\), \(p(\theta| u)\), or when one wants to obtain a point estimate \(\theta^\star = \argmax_\theta (p(\theta | u))\) (frequentist or machine learning context), the model gradient proves very useful. In a full Bayesian inference context, they are used e.g. with Hamiltonian Markov Chains methods, such as the NUTS sampler, and in a machine learning context, they are used with gradient-based optimizer.
Gradient descent
The best way to grasp the importance of gradients in inference is to understand the gradient descent algorithm.
The following picture illustrates the algorithm in the special case where \(p\) is one-dimensional.
Given an initial estimate of the parameter value \(p_0\), \(\frac{d \mathcal{M}}{dp}\) is used to suggest a new, better estimate, following
\[p_{n+1} = p_n - \eta \frac{d \mathcal{M}}{dp}(u_0, t, p) \]
where \(\eta\) is the learning rate.
Gradient-based methods are usually very efficient in high-dimensional spaces.
Automatic differentiation
Let’s go back to our method ∂mymodel_∂p. What is the optimal value of h to calculate the derivative? This is a tricky question, because a too small h can lead to round off errors (see more explanations here) while h too large also leads to a bad approximation of the asymptotic definition.
Fortunately, a bunch of techniques referred to as automatic differentiation (AD) allows to exactly differentiate any piece of numerical functions. In practice, your code must be exclusively written within an AD-backend, such as Torch, JAX or Tensorflow. Those libraries do not know how to differentiate code not written in their own language, such as normal Python code.
Fortunately, Julia is an AD-pervasive language! This means that any piece of Julia code is theoretically differentiable with AD.
using ForwardDiff
@btime ForwardDiff.gradient(p -> mymodel(u0, p, 1.), p); 1.225 μs (12 allocations: 432 bytes)This is what makes Julia great for model calibration and inference! Write your model in Julia, and any inference method using AD will be able to work with your model!
To learn more about AD in Julia, check-out this cool blog-post and this short presentation.
Now let’s get started with inference.
Mechanistic inference
The mechanistic model and the data
We’ll use a simple dynamical community model, the Lotka Volterra model, to generate data. We’ll then contaminate this data with noise, and try to recover the parameters that have generated the data. The goal of the session will be to estimate those parameters from the data, using a bunch of different techniques.
So let’s first generate the data.
using OrdinaryDiffEq
# Define Lotka-Volterra model.
function lotka_volterra(du, u, p, t)
# Model parameters.
@unpack α, β, γ, δ = p
# Current state.
x, y = u
# Evaluate differential equations.
du[1] = (α - β * y) * x # prey
du[2] = (δ * x - γ) * y # predator
return nothing
end
# Define initial-value problem.
u0 = [2.0, 2.0]
p_true = (;α = 1.5, β = 1.0, γ = 3.0, δ = 1.0)
# tspan = (hudson_bay_data[1,:t], hudson_bay_data[end,:t])
tspan = (0., 5.)
tsteps = range(tspan[1], tspan[end], 51)
alg = Tsit5()
prob = ODEProblem(lotka_volterra, u0, tspan, p_true)
saveat = tsteps
sol_true = solve(prob, alg; saveat)
# Plot simulation.
plot(sol_true)This is the true state of the system. Now let’s contaminate it with observational noise.
Now that we have our data, let’s do some inference!
Mechanistic inference as a supervised learning task
We’ll get started with a very crude approach to inference, where we’ll treat the calibration of our LV model similarly to a supervised machine learning task. To do so, we’ll write a loss function, defining a distance between our model and the data, and we’ll try to minimize this loss. The parameter minimizing this loss will be our best model parameter estimate.
function loss(p)
predicted = solve(prob,
alg;
p,
saveat,
abstol=1e-6,
reltol = 1e-6)
l = 0.
for i in 1:length(predicted)
if all(predicted[i] .> 0)
l += sum(abs2, log.(data_mat[:, i]) - log.(predicted[i]))
end
end
return l, predicted
endloss (generic function with 1 method)
Note
Notice that we explicitly check whether predictions are > 0, because the log of negative number is not defined and will throw an error!
Let’s define a helper function, that will plot how good does the model perform across different iterations.
losses = []
callback = function (p, l, pred; doplot=true)
push!(losses, l)
if length(losses)%100==1
println("Current loss after $(length(losses)) iterations: $(losses[end])")
if doplot
plt = scatter(tsteps, data_mat', color = [1 2], label=["Prey abundance data" "Predator abundance data"])
plot!(plt, tsteps, pred', color = [1 2], label=["Inferred prey abundance" "Inferred predator abundance"])
display(plot(plt, yaxis = :log10, title="it. : $(length(losses))"))
end
end
return false
end#13 (generic function with 1 method)And let’s define a wrong initial guess for the parameters
pinit = ComponentArray(;α = 1., β = 1.5, γ = 1.0, δ = 0.5)
callback(pinit, loss(pinit)...; doplot = true)Current loss after 1 iterations: 251.10349846646116falseOur initial predictions are bad, but you’ll likely get even worse predictions in a real-case scenario!
We’ll use the library Optimization, which is a wrapper library around many optimization libraries in Julia. Optimization therefore provides us with many different types of optimizers to find parameters minimizing loss. We’ll specifically use the infamous Adam optimizer (187k citations!!!), widely used in ML.
using Optimization
using OptimizationOptimisers
using SciMLSensitivity
adtype = Optimization.AutoZygote()
optf = Optimization.OptimizationFunction((x, p) -> loss(x), adtype)
optprob = Optimization.OptimizationProblem(optf, pinit)
@time res_ada = Optimization.solve(optprob, Adam(0.1); callback, maxiters = 500)
res_ada.minimizerCurrent loss after 101 iterations: 8.039887486778179Current loss after 201 iterations: 7.9094080306025445Current loss after 301 iterations: 7.806219868794404Current loss after 401 iterations: 7.74345616951535Current loss after 501 iterations: 7.712910946192632 13.323594 seconds (49.62 M allocations: 3.144 GiB, 3.72% gc time, 93.71% compilation time: 9% of which was recompilation)ComponentVector{Float64}(α = 1.5322556800023097, β = 1.0159023620691514, γ = 2.8926590524331766, δ = 0.9148575218436299)
Nice! It seems that the optimizer did a reasonable job, and that we found a reasonable estimate of our parameters!
Regularization
In supervised learning, it is common practice to regularize the model to prevent overfitting. Regularization can also help the model to converge. Regularization is done by adding a penalty term to the loss function:
\[ \text{Loss}(\theta) = \text{Loss}_{\text{data}}(\theta) + \lambda \, \text{Reg}(\theta) \]
Multiple shooting
Another trick that can greatly improve the convergence of the optimization is to break down the prediction task into simpler tasks. Namely, instead of trying to predict in one shot the whole time series, the idea of multiple shooting is to predict for shorter time horizon.
Sensitivity methods
Did you wonder why do we need to load SciMLSensitivity? and why did we specify adtype = Optimization.AutoZygote()?
AD comes in different flavours, with broadly two types of AD methods - forward methods and backward methods -, and a bunch of different implementations.
You can specify which ones Optimization.jl will use to differentiate loss with adtype, see available options here.
But when it comes to differentiating the solve function from OrdinaryDiffEq, you want to use AutoZygote(), because when trying to differentiate solve, a specific adjoint rule provided by the SciMLSensitivity package will be used.
What are adjoint rules?
These are algoirithmic rules that specify to the AD backend how to calculate the derivative of a specific function.
If you want to know more, check-out the ChainRules.jl documentation
These adjoint rules can be specificed by the keyword sensealg when calling solve and have been designed for best performance when differentiating solutions of an ODEProblem. There exists a lot of them (see a review here), and if sensealg is not provided, a smart polyalgorithm is going to pick up one for you.
You can have a look in the documentation here for hints on how to choose an algorithm.
For a small number of parameters, forward methods tend to perform best, but with higher number of parameters, the other way around is true.
Well done! Now, let’s jump into the Bayesian world…
Bayesian inference
Julia has a very strong library for Bayesian inference: Turing.jl.
Let’s declare our first Turing model!
This is done with the @model macro, which allows the library to automatically construct the posterior distribution based on the definition of your model’s random variables.
Frequentist (supervised learning) vs. Bayesian approach
The main difference between a frequentist approach and a Bayesian approach is that the latter considers that parameters are random variables. Hence instead of trying to estimate a single value for the parameters, the Bayesian will try to estimate the posterior (joint) distribution of those parameters.
\[ P(\theta | \mathcal{D}) = \frac{P(\mathcal{D} | \theta) P(\theta)}{P(\mathcal{D})} \]
Random variables are defined with the ~ symbol.
Our first Turing model
using Turing
using LinearAlgebra
@model function fitlv(data, prob)
# Prior distributions.
σ ~ InverseGamma(3, 0.5)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.2, 0.5); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.5); lower=0, upper=2)
# Simulate Lotka-Volterra model.
p = (;α, β, γ, δ)
predicted = solve(prob, alg; p, saveat)
# Observations.
for i in 1:length(predicted)
if all(predicted[i] .> 0)
data[:, i] ~ MvLogNormal(log.(predicted[i]), σ^2 * I)
end
end
return nothing
endfitlv (generic function with 2 methods)Now we can instantiate our model, and run the inference!
model = fitlv(data_mat, prob)
# Sample 3 independent chains with forward-mode automatic differentiation (the default).
chain = sample(model, NUTS(), MCMCThreads(), 1000, 3; progress=true)Chains MCMC chain (1000×17×3 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 3
Samples per chain = 1000
Wall duration = 25.55 seconds
Compute duration = 24.3 seconds
parameters = σ, α, β, γ, δ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
σ 0.2796 0.0195 0.0005 1721.3574 1789.3390 1.0013 ⋯
α 1.4928 0.1501 0.0052 841.9730 862.7118 1.0025 ⋯
β 0.9902 0.1210 0.0040 907.2968 995.0953 1.0008 ⋯
γ 2.9967 0.2656 0.0090 863.2374 992.8786 1.0045 ⋯
δ 0.9592 0.1043 0.0034 939.4457 1141.4992 1.0034 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ 0.2449 0.2656 0.2788 0.2922 0.3212
α 1.2283 1.3875 1.4784 1.5863 1.8276
β 0.7748 0.9077 0.9816 1.0661 1.2575
γ 2.4786 2.8192 2.9973 3.1727 3.5358
δ 0.7563 0.8869 0.9584 1.0274 1.1650
Threads
How many threads do you have running? Threads.nthreads() will tell you!
Let’s see if our chains have converged.
using StatsPlots
plot(chain)Data retrodiction
Let’s now generate simulated data using samples from the posterior distribution, and compare to the original data.
function plot_predictions(chain, sol, data_mat)
myplot = plot(; legend=false)
posterior_samples = sample(chain[[:α, :β, :γ, :δ]], 300; replace=false)
for parr in eachrow(Array(posterior_samples))
p = NamedTuple([:α, :β, :γ, :δ] .=> parr)
sol_p = solve(prob, Tsit5(); p, saveat)
plot!(sol_p; alpha=0.1, color="#BBBBBB")
end
# Plot simulation and noisy observations.
plot!(sol; color=[1 2], linewidth=1)
scatter!(sol.t, data_mat'; color=[1 2])
return myplot
end
plot_predictions(chain, sol_true, data_mat)Here is a small utility function to visualize your results.
plot_predictions2
function plot_predictions2(chain, sol, data_mat)
myplot = plot(; legend=false)
posterior_samples = sample(chain, 300; replace=false)
for i in 1:length(posterior_samples)
ps = posterior_samples[i]
p = get(ps, [:α, :β, :γ, :δ], flatten=true)
u0 = get(ps, :u0, flatten = true)
u0 = [u0[1][1], u0[2][1]]
sol_p = solve(prob, Tsit5(); u0, p, saveat)
plot!(sol_p; alpha=0.1, color="#BBBBBB")
end
# Plot simulation and noisy observations.
plot!(sol; color=[1 2], linewidth=1)
scatter!(sol.t, data_mat'; color=[1 2])
return myplot
end
plot_predictions2(chain2, sol_true, data_mat)Mode estimation
Turing allows you to find the maximum likelihood estimate (MLE) or maximum a posteriori estimate (MAP).
\[ \theta_{MLE} = \underset{\theta}{\text{argmax}} \, P(\mathcal{D} | \theta), \qquad \theta_{MAP} = \underset{\theta}{\text{argmax}} \, P(\theta | \mathcal{D}). \]
MAP and regularization in supervised learning
Although Bayesian inference seems very different from the supervised learning approach we developed in the first part, estimating the MAP, which can be still considered as Bayesian inference, transforms in an optimization problem that can be seen as a supervised task.
To see that, we can log-transform the posterior:
\[ \log P(\theta | \mathcal{D}) = \log P(\mathcal{D} | \theta) + \log P(\theta) - \log P(\mathcal{D}) \]
Since the evidence \(P(\mathcal{D})\) is independent of \(\theta\), it can be ignored when maximizing with respect to \(\theta\). Therefore, the MAP estimate simplifies to:
\[ \theta_{MAP} = \underset{\theta}{\text{argmax}} \, \left[\log P(\mathcal{D} | \theta) + \log P(\theta)\right] \]
Here, \(\log P(\mathcal{D}| \theta)\) can be seen as our previous non-regularized loss and \(\log P(\theta)\) acts as a regularization term, penalizing unlikely parameter values based on our prior beliefs. Priors on parameters can be seen as regularization term.
MLE and MAP can be obtained by maximum_likelihood and maximum_a_posteriori.
Random.seed!(0)
maximum_a_posteriori(model2, maxiters = 1000)ModeResult with maximized lp of -104.88
[0.3545376205457767, 1.4695692517420373, 0.9162499950736273, 3.263944963496157, 1.0243607922108577, 2.150749205538098, 2.4795481828054595]Since Turing uses under the hood the same Optimization.jl library, you can specify which optimizer youd’d like to use.
map_res = maximum_a_posteriori(model2, Adam(0.01), maxiters=2000)ModeResult with maximized lp of -104.88
[0.35455374965749115, 1.4707686527453756, 0.9171941147556801, 3.2614628620071664, 1.0235193248242322, 2.1506473758409883, 2.4789084651090993]We can check whether the optimization has converged:
@show map_res.optim_resultmap_res.optim_result = retcode: Default
u: [-1.036895323466616, -0.05847935462336067, -0.16599185850450063, 1.1190957778225292, 0.04704732580671333, 0.765768901858866, 0.9078183282523818]
Final objective value: 104.87762402604213
retcode: Default
u: 7-element Vector{Float64}:
-1.036895323466616
-0.05847935462336067
-0.16599185850450063
1.1190957778225292
0.04704732580671333
0.765768901858866
0.9078183282523818What’s very nice is that Turing.jl provides you with utility functions to analyse your mode estimation results.
using StatsBase
coeftable(map_res)| Coef. | Std. Error | z | Pr(> | z | ) | |
|---|---|---|---|---|---|---|
| σ | 0.354554 | 0.0250558 | 14.1506 | 1.85249e-45 | 0.305445 | 0.403662 |
| α | 1.47077 | 0.157711 | 9.32571 | 1.10241e-20 | 1.16166 | 1.77988 |
| β | 0.917194 | 0.125103 | 7.3315 | 2.27594e-13 | 0.671996 | 1.16239 |
| γ | 3.26146 | 0.335101 | 9.73279 | 2.18526e-22 | 2.60468 | 3.91825 |
| δ | 1.02352 | 0.124744 | 8.20497 | 2.30644e-16 | 0.779026 | 1.26801 |
| u0[1] | 2.15065 | 0.18462 | 11.6491 | 2.31953e-31 | 1.7888 | 2.51249 |
| u0[2] | 2.47891 | 0.247352 | 10.0218 | 1.22272e-23 | 1.99411 | 2.96371 |
AD backends and sensealg
The NUTS sampler uses automatic differentiation under the hood.
By default, Turing.jl uses ForwardDiff.jl as an AD backend, meaning that the SciML sensitivity methods are not used when the solve function is called. However, you could change the AD backend to Zygote with adtype=AutoZygote().
chain2 = sample(model2, NUTS(), MCMCThreads(), adtype=AutoZygote(), 3000, 3; progress=true)Chains MCMC chain (3000×19×3 Array{Float64, 3}):
Iterations = 1001:1:4000
Number of chains = 3
Samples per chain = 3000
Wall duration = 57.4 seconds
Compute duration = 56.91 seconds
parameters = σ, α, β, γ, δ, u0[1], u0[2]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
σ 0.3690 0.0267 0.0003 6634.3954 6080.5757 1.0000 ⋯
α 1.5026 0.1596 0.0030 2825.0996 3566.4279 1.0004 ⋯
β 0.9458 0.1303 0.0024 3055.7314 3652.9872 1.0015 ⋯
γ 3.2448 0.3214 0.0060 2806.7123 2850.3476 1.0009 ⋯
δ 1.0199 0.1212 0.0022 3167.1794 3679.8852 1.0008 ⋯
u0[1] 2.1903 0.2017 0.0026 6066.7978 5252.2598 1.0001 ⋯
u0[2] 2.4814 0.2547 0.0034 5638.9388 5057.7901 1.0007 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ 0.3219 0.3507 0.3675 0.3854 0.4254
α 1.2290 1.3890 1.4889 1.6003 1.8558
β 0.7245 0.8536 0.9332 1.0234 1.2380
γ 2.6101 3.0243 3.2492 3.4636 3.8759
δ 0.7863 0.9365 1.0169 1.1007 1.2611
u0[1] 1.8288 2.0508 2.1789 2.3149 2.6257
u0[2] 2.0080 2.3053 2.4727 2.6454 3.0176
Doing so, you could specify within solve the adtype. It is usually a good idea to try a few different sensitivity algorithm.
See here for more information.
Variational Inference
Variational inference (VI) consists in approximating the true posterior distribution \(P(\theta | \mathcal{D})\) by an approximate distribution \(Q(\theta; \phi)\), where \(\phi\) is a parameter vector defining the shape, location, and other characteristics of the approximate distribution \(Q\), to be optimzed so that \(Q\) is as close as possible to \(P\). This is achieved by minimizing the Kullback-Leibler (KL) divergence between the true posterior \(P(\theta | \mathcal{D})\) and the approximate distribution :
\[ \phi^* = \underset{\phi}{\text{argmin}} \, \text{KL}\left(Q(\theta; \phi) \,||\, P(\theta | \mathcal{D})\right) \]
The advantage of VI over traditional MCMC sampling methods is that VI is generally faster and more scalable to large datasets, as it transforms the inference problem into an optimization problem.
Let’s do VI in Turing!
import Flux
using Turing: Variational
model = fitlv2(data_mat, prob)
q0 = Variational.meanfield(model)
advi = ADVI(10, 10_000) # first arg is the
q = vi(model, advi, q0; optimizer=Flux.ADAM(1e-2))
function plot_predictions_vi(q, sol, data_mat)
myplot = plot(; legend=false)
z = rand(q, 300)
for parr in eachcol(z)
p = NamedTuple([:α, :β, :γ, :δ] .=> parr[2:5])
u0 = parr[6:7]
sol_p = solve(prob, Tsit5(); u0, p, saveat)
plot!(sol_p; alpha=0.1, color="#BBBBBB")
end
# Plot simulation and noisy observations.
plot!(sol; color=[1 2], linewidth=1)
scatter!(sol.t, data_mat'; color=[1 2])
return myplot
end
plot_predictions_vi(q, sol_true, data_mat)The cool thing with VI that we can sample from the resulting q with ease.
q isa MultivariateDistributiontruerand(q)7-element Vector{Float64}:
0.3910702850249754
1.8261988965103004
1.169798842596696
2.80613428184438
0.8402820867003005
2.313112765625009
2.406422925384114Infering functional forms
Up to now, we have been infering the value of the model’s parameters, assuming that the structure of our model is correct. But this is very idealistic, specifically in ecology. As a general trend, we have little idea of how does e.g. the functional response of a species look like.
What if instead of inferring parameter values, we could infer functional forms, or components within our model for which we have little idea on how to express it mathematically?
In Julia, we can do that.
To illustrate this, we’ll assume that we do not know the functional response of both prey and predator, i.e. the terms β * y and δ * x. Instead, we will parametrize this component in our DE model by a neural network, which can be seen as a simple non-linear regressor dependent on some extra parameters p_nn.
We then simply have to optimize those parameters, along with the other model’s parameters!
Let’s get started. To make the neural network, we’ll use the deep learning library Lux.jl, which is similar to Flux.jl but where models are explicitly parametrized. This explicit parametrization makes it simpler to integrate with an ODE model.
To make things simpler, we will define a single layer neural network
using Lux
Random.seed!(2)
rng = Random.default_rng()
nn_init = Lux.Chain(Lux.Dense(2,2, relu))
p_nn_init, st_nn = Lux.setup(rng, nn_init)
nn = StatefulLuxLayer(nn_init, st_nn)StatefulLuxLayer{true}(
Dense(2 => 2, relu), # 6 parameters
) # Total: 6 parameters,
# plus 0 states.
We use a StatefulLuxLayer to not having to carry around st_nn, a struct containing states of a Lux model, which is essentially useless for a multi-layer perceptron.
st_nnNamedTuple()We can now evaluate our neural network model as follows:
nn(u0, p_nn_init)2-element Vector{Float64}:
0.0
0.0instead of
nn_init(u0, p_nn_init, st_nn)([0.0, 0.0], NamedTuple())Let’s define a new parameter vectors, which will consist of the ODE model parameters as well as the neural net parameters
pinit = ComponentArray(;σ = 0.3, α = 1., γ = 1.0, p_nn=p_nn_init)ComponentVector{Float64}(σ = 0.3, α = 1.0, γ = 1.0, p_nn = (weight = [-1.0083649158477783 -0.7284937500953674; -1.219232201576233 0.4427390396595001], bias = [0.0; 0.0;;]))
Let’s check our initial model predictions:
prob_nn = ODEProblem(lotka_volterra_nn, u0, tspan, pinit)
init_sol = solve(prob_nn, alg; saveat)
# Plot simulation.
plot(init_sol)Now we can define our Turing Model. We’ll need to use a utility function vector_to_parameters that reconstructs the neural network parameter type based on a sampled parameter vector (taken from this Turing tutorial). You do not need to worry about this. Note that we could have used a component vector, but for some reason this did not work at the time of the writing of this tutorial…
vector_to_parameters
using Functors # for the `fmap`
function vector_to_parameters(ps_new::AbstractVector, ps::NamedTuple)
@assert length(ps_new) == Lux.parameterlength(ps)
i = 1
function get_ps(x)
z = reshape(view(ps_new, i:(i + length(x) - 1)), size(x))
i += length(x)
return z
end
return fmap(get_ps, ps)
endvector_to_parameters (generic function with 1 method)# Create a regularization term and a Gaussian prior variance term.
sigma = 0.2
@model function fitlv_nn(data, prob)
# Prior distributions.
σ ~ InverseGamma(3, 0.5)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
nparameters = Lux.parameterlength(nn)
p_nn_vec ~ MvNormal(zeros(nparameters), sigma^2 * I)
p_nn = vector_to_parameters(p_nn_vec, p_nn_init)
# Simulate Lotka-Volterra model.
p = (;α, γ, p_nn)
predicted = solve(prob, alg; p, saveat)
# Observations.
for i in 1:length(predicted)
if all(predicted[i] .> 0)
data[:, i] ~ MvLogNormal(log.(predicted[i]), σ^2 * I)
end
end
return nothing
end
model = fitlv_nn(data_mat, prob_nn)DynamicPPL.Model{typeof(fitlv_nn), (:data, :prob), (), (), Tuple{Matrix{Float64}, ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, ComponentVector{Float64, Vector{Float64}, Tuple{Axis{(σ = 1, α = 2, γ = 3, p_nn = ViewAxis(4:9, Axis(weight = ViewAxis(1:4, ShapedAxis((2, 2))), bias = ViewAxis(5:6, ShapedAxis((2, 1))))))}}}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(lotka_volterra_nn), UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}}, Tuple{}, DynamicPPL.DefaultContext}(fitlv_nn, (data = [1.8655845948955276 2.298199048573464 … 4.071164055293614 5.672667515002083; 2.651867857608795 3.2812317734519048 … 1.351784872962806 1.1243450946947573], prob = ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, ComponentVector{Float64, Vector{Float64}, Tuple{Axis{(σ = 1, α = 2, γ = 3, p_nn = ViewAxis(4:9, Axis(weight = ViewAxis(1:4, ShapedAxis((2, 2))), bias = ViewAxis(5:6, ShapedAxis((2, 1))))))}}}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(lotka_volterra_nn), UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}(ODEFunction{true, SciMLBase.AutoSpecialize, typeof(lotka_volterra_nn), UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}(lotka_volterra_nn, UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing, nothing, nothing), [2.0, 2.0], (0.0, 5.0), (σ = 0.3, α = 1.0, γ = 1.0, p_nn = (weight = [-1.0083649158477783 -0.7284937500953674; -1.219232201576233 0.4427390396595001], bias = [0.0; 0.0;;])), Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}(), SciMLBase.StandardODEProblem())), NamedTuple(), DynamicPPL.DefaultContext())using Optimization, OptimizationOptimisers
@time map_res = maximum_a_posteriori(model, ADAM(0.05), maxiters=3000, initial_params=pinit)
pmap = ComponentArray(;σ=0, pinit...)
pmap .= map_res.values[:]
sol_map = solve(prob_nn, alg;p=pmap, saveat, tspan = (0, 10))
scatter(tsteps, data_mat', color = [1 2], label=["Predator abundance data" "Prey abundance data"])
plot!(sol_map, color = [1 2], label=["Inferred predator abundance" "Inferred prey abundance"], yscale=:log10) 11.941995 seconds (31.83 M allocations: 5.634 GiB, 3.35% gc time, 86.92% compilation time: <1% of which was recompilation)This seems to fail.
Happy with the convergence? Now let’s investigate what did the neural network learn!
plot_func_resp
function plot_func_resp(p, data)
# plotting prediction of functional response
u1 = range(minimum(data[1,:]), maximum(data[1,:]), length=100)
u2 = range(minimum(data[2,:]), maximum(data[2,:]), length=100)
u = hcat(u1,u2)
func_resp = nn(u', p.p_nn)
myplot1 = plot(u2,
- p_true.β .* u2;
label="True functional form",
xlabel="Predator abundance")
plot!(myplot1,
u2,
- func_resp[1,:];
color="#BBBBBB",
label="Inferred functional form")
myplot2 = plot(u1,
p_true.δ .* u2;
legend=false, xlabel="Prey abundance")
plot!(myplot2,
u1,
func_resp[2,:];
color="#BBBBBB")
myplot = plot(myplot1, myplot2)
return myplot
endplot_func_resp (generic function with 1 method)plot_func_resp(pmap, data_mat)The neural network has well captured the functional forms of the predator and prey.
This concludes this tutorial; I hope that it has given you plenty of ideas for your future research projects!
Resources
- https://turinglang.org/docs/tutorials/10-bayesian-differential-equations/
- https://turinglang.org/docs/tutorials/09-variational-inference/